discrete probability distributions farrokh alemi ph.d

DISCRETE PROBABILITY DISTRIBUTIONS

Farrokh Alemi Ph.D.

BernoulliGeometricBinomialPoisson

Discrete Probability Distributions

Density function

Definitions

Cumulative probability function

Definitions

Definitions

Expected Value

Expected Value

Expected Value

Expected Value

Example

Example

Example

Example

10Otherwise

10.042 medication errors

0.960.221 medication error

0.740.740 medication errors

Cumulative distribution

function

Probability density functionEvents

10Otherwise

10.042 medication errors

0.960.221 medication error

0.740.740 medication errors

Cumulative distribution

function

Probability density functionEvents

Example

Example

Density & Cumulative Distributions

BernoulliGeometricBinomialPoisson

Typical Probability Density Functions

Mutually exclusive

Bernoulli Probability Density Function

Exhaustive

Bernoulli Probability Density Function

Bernoulli Probability Density Function

pOccurs

1-PDoes not occur

Probability density functionEvents

pOccurs

1-PDoes not occur

Probability density functionEvents

Bernoulli Probability Density Function

Independent Bernoulli Trials

Independent Bernoulli Trials

Geometric Density Function

Geometric Density Function

Geometric Density Function

No medication errors have occurred in the past 90 days. What is the maximum daily probability of medication error in our facility?

Do One

The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls?

Do One

Number of k occurrences of the event in n independent trials

Binomial Probability Distribution

Independent Bernoulli Trials

Independent Bernoulli Trials

Independent Bernoulli Trials

Independent Bernoulli Trials

P x (1-P) x P

Different combinationsSuccess probabilitiesFailure probabilities

Binominal Probability Distribution

Binomial Probability Distribution

Binomial Probability Distribution

Binomial Probability Distribution

6 Trials of Binomial p=1/2

6 Trials of Binomial p=1/2

6 Trials of Binomial p=0.05

If the monthly probability of elopement is 0.05, how many patients will elope in 2 years?

Example

If the monthly probability of elopement is 0.05, how many patients will elope in 2 years?

Example

If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days?

Example

If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days?

Example

Which is more likely, 2 patients failing to comply with medication orders in 15 days or 3 patients failing to comply with medication orders in 30 days.

Do One

Large number of trialsSmall probabilities of occurrence

Poisson Density Function

Poisson Density Function

k

Poisson Density Function

k

Repeated independent Bernoulli trials is the foundation of many distributions

Take Home Lesson

What is the probability of observing one or more security violations, when the daily probability of violations is .01 and we are monitoring the organization for 4 months.

Do One

How many visits will it take to have at least one medication error if the estimated probability of medication error in a visit is 0.03?

Do Another